Relational structures play key role in mathematics. Partial orders, chains, graphs, trees etc. are some of significant relational structures. Given a relational structure E, the sibling number of E is the number of relational structures mutually embeddable and non-isomorphic to E. Thomassé's Conjecture states that the sibling number of any countable relational structure is either 1 or $\aleph_0$ or $2^{\aleph_0}$. There is an alternative form of this conjecture in which for a relational structure the sibling number is either 1 or $\infty$. The conjectures have been specifically verified for some important structures such as rayless graphs, chains, $\aleph_0$-categorical structures, cographs etc. The conjectures are still open in general. We present a history and significant results related to them. We also give a survey of the conjecture about locally finite trees and direct sum of chains.